Swelling of Macromolecules Grafted Onto an Inert Surface 141
Vol. 12, No. I , January-February 1979
Swelling of Macromolecules Grafted on Inert Surfaces and Partitioning of a Solute between Solvent and Grafted Phase J a c q u e l i n e Lecourtier, Roland Audebert,* a n d C l a u d e Quivoron Laboratoire de Physico-Chimie Macromol&culaire, Universite Pierre et Marie Curie, Associ6 a u C.N.R.S., 75231 Paris, Cedex 05, France. Received February 8, 1978 ABSTRACT: A method is proposed for determining the thermodynamic characteristics of macromolecules
grafted on an inert plane. Based on Flory's theory for macromolecular solutions and gels, the swelling of such a grafted phase by a solvent is calculated. Results are applied to evaluate the partition coefficient of a solute between grafted molecules and a solvent. Variations of this coefficient with the molecular sizes of the solvent, solute, and grafts and with the interactions between these molecules are studied.
T h e behavior of a macromolecule in the vicinity of an interface is useful for understanding the adsorption of p o l y m e r ~ l -and ~ for explaining the fractionation observed in gel permeation chromatography (GPC).4-6 In the particular case of a macromolecule attached by one of its ends to a surface, numerous applications are suggested: liquid-liquid chromatography, selective absorption of pollutants, compatibilization of prosthesis with blood, stabilization of polymer emulsions, supported catalysis, etc. The thickness of the polymer layer adsorbed on a surface and its variation with molecular weight has been experimentally The segmental distribution of grafted macromolecules has been evaluated both theoretically and by ~ i m u l a t i o n but ~ ~ ' their ~ thermodynamic characteristics have not been systematically studied. These characteristics should allow one to better understand the behavior of grafted materials. Hence, we investigated the problem and using Flory's theory on macromolecular solutions,13J4we calculated the swelling by a solvent of a macromolecular phase grafted on a plane and we deduced the partition coefficient of a solute between the two phases of such a system.15 Model As depicted in Figure 1, we take as our model N 3 nonentangling polymer chains grafted onto an inert plane. That is, chains are independent and there is no adsorption effect on the wall. According to the lattice liquid theory13J4 solvent and polymer molecules are assumed to be constituted of segments of identical volume. Thus, each molecule of solvent is composed of rl segments and each molecule of polymer is composed of r3 segments. This rough model, representative of graft phases used in practice, requires the specification of only a few parameters and the resulting expressions can be easily applied to experimental data. For example, it elucidates the behavior of chemically bonded stationary phases which have gained increasing importance in liquid chromatography.
Free E n e r g y of M i x t u r e s The free-energy change involved in the mixing of a solvent with the grafted phase is AF IF = in which
AFM
AFM
+ AFEL
(la)
1. Computing E n t r o p y of A S M Mix. Consider a solution consisting of N , molecules of solvent and N3 molecules of polymer. The number of permissible patterns of the two kinds of molecules in such a system is Q. When the polymer molecules have one end attached to a surface, the pattern number becomes Q'. Q' is smaller than Q: Q' = QPP'
(2)
P is the fraction of all the configurations that would still remain available if the position of one end of all the polymer molecules were fixed in the solution. P' is the fraction of these permissible configurations when all such fixed points are on a plane bounding the solution. The symbols Q',,, Qo, Po, and P i denote the corresponding quantities for pure compounds. a. Calculating P . Using a lattice model, any solution of N 3 polymer molecules (r3 segments) and N1 solvent cells. We molecules (rl segments) comprises Nl (r3/rl)N3 consider all the permissible configurations of this system. We can choose N 3 cells to be occupied by one polymer chain end. For each specific choice of N3 cells, the number of permissible configurations is the same. Hence for a given choice of N3 cells, the corresponding fraction of the total number of configurations is:
+
For our purpose, a solution of N 3 one-end-bonded macromolecules and N1 solvent molecules corresponds to a particular choice of N 3 cells. Thus P is derived from eq 3. It follows that for pure compounds:
b. Calculating P'. The fraction p : of all conformatins of a polymer chain ( r 3 segments) that are still available when one end of the chain is fixed a t a distance x of an impenetrable plane boundary has been evaluated by Casassa16
is the free energy of mixing AFM = A H M
-
TASM
(Ib)
and A F E L is the elastic free energy due to the molecular expansion of the grafted chains in the solvent.
in which b is the length of one of the r 3 / r 1 units of the polymer chain.
0024-9297/79/2212-0141$01.00/0 t S 1979 American Chemical Society
142 Lecourtier, Audebert, Quivoron
Macromolecules conformation, following a deformation which is characterized by an expansion parameter a and is constrained in the following ways: the ends of N3,ichains are within a differential volume element Ax AyAz located about point (xi,yi,zi);a chain end having components (xi/a,yi/a,zi/a) has components (xi,yi,zi) after deformation. When the deformation process is swelling, cy3 = l / $ in which $3 is the volume fraction of the polymer in the swollen phase. The probability that any given grafted chain has components (zi,yi,zi)within the range (Ax,Ay,Az) is:
wi = W(xi,yi,zi)Ax AyAz in which Figure 1. Depiction of grafted chains using a lattice model.
The flexible portion of a polymer chain terminally attached to a plane begins a t a distance b from the plane. Thus we shall consider that in our case x = b, after which applying eq 5 gives P'b,o, i.e.,
and
Flory's calculation for N 3 chains leads to: When the attached molecule is swollen by a solvent, its volume fraction is $3 and the average distance between two points of the chain is multiplied by $3-1/3. As the pinning distance is unmodified, Casassa's result leads to:
in which N3,iis given bf
xi yi zi AXAYAZ N3,i(xi,yi,zi)= N3w -,-,a3
Taking logarithms and introducing Stirling's approximation for the factorials: Here, Pb,o and P b , l refer to one molecule. When N 3 identical but non-entangling macromolecules are grafted, the total number of configurations is:
R' = fiP(p'b,1)N3= QPP' c. Expression of the Entropy of Mixing ASM. The expression of the entropy of mixing is:
R'
Substituting the summation by appropriate integrations we obtain In Rar =
P
ASM = K In 7= K a0
According to Flory's lattice theory of polymer solutions,13J4the term In (R/Ro)is expressed by In (R/Ro)= -(Nl In $1 N 3 In $3). Substituting Stirling's approximations for the factorials appearing in the expressions for P and p b , l leads after some simplification to:
+
ASM
= -N1 In Y 11
[
$1
+
N 3 ( ; - 1)
+ Nl]
[p2(1/a2- 1)(x2
+ y2 + z2) + In a3]dx dy dz
(14)
i.e.,
1 The entropy change involved in deformation is obtained by substracting from k In REL the value of S for a3 = 1:
In (1
-
;$3)
-
2. Evaluating A S E L . By analogy with Flory's theory on ge1s,13,14we must find the probability REL that the N3 grafted chains will occur spontaneously in a specified
3. Computing Term AHhl. We can assume that the interaction energy between a polymer segment and a solvent molecule remains unchanged after grafting. Hence, A H M is the classical enthalpy of mixing calculated by F10ry:'~J~ m h l
= kTX13N1(b3
(16)
Swelling of Macromolecules Grafted Onto an Inert Surface 143
Vol. 12, No. 1, January-February 1979
change A F involved in the mixing of a solute with the swollen grafted phase: AFM =
L W M-
T ( A S M+ ASEL)
in which aHM
= X13N143 + xl2Nl42+
X23r3N342
(19)
and
Figure 2. Swelling of grafted phase (43) vs. xi3 for various values of r3 and by taking r1 = 1.
in which x i 3 is Flory’s interaction parameter. 4. Evaluating the Swelling. According to the previous equations the expression of AFM is:
AFM = k T [ N 1 In Cpl -
5d3) + N3(; 73.
[
x12is Flory’s solvent/solute interaction parameter and ~ 2 3 represents the interaction intensity between a segment of the grafted chains and a segment of the solute. 2. Computing A S w According to Flory’s theory,13J4 if the ternary system were composed of N 1 solvent molecules, N 2 solute molecules, and N3 chains in solution, the total number of configurations of this system would be In ( 9 / Q 2 ,=) - [ N , In 4,
N1 + N3(:
-
1) In (1 -
-
l ) ] In (1 -
2)
-
+ N 2 In d2 + N3 In 431
(21)
I t has been shown above that fl becomes 9’ when the N 3 polymer molecules are actually grafted on a plane: 9‘ = 9PP’
Disregarding the steric constraint of the wall, the distribution of the solute in the grafted phase would be homogeneous. Assuming the grafted phase to be a layer of a swollen polymer of thickness e, the probability of finding one end of a solute molecule between distances x and x + dx from the wall is d9’ = 9 dxle. In fact the vicinity of the wall decreases this number of permissible configurations by the factor p .: According to Casassa’s calculation:
Hence solvent activity in the swollen phase is:
It follows that:
Assuming thickness e of the grafted phase to be the diameter of the equivalent sphere of a dissolved macromolecule having the same swelling as the grafted macromolecule, then e = a( ?)li2b
Extenting the steric effect of the wall to all the grafted molecules, the total number of permissible arrangements 9” of the solute becomes
At the swelling equilibrium, In a, = 0. In Figure 2, xI3 is plotted against 93 for each value of r3 (rl = 1). We mention that the swelling of the grafted phase increases with r3 and decreases with ~ 1 3 .
Calculation of the Partition Coefficient KD of a Macromolecular Solute between a Grafted Polymer and a Solvent 1. Free-Energy Formula. Extension of the previous equations to a system of three components, using r2,for example, to characterize the solute, leads to the free-energy
a(r3/ r l )l/zb
After integration one finds
= ~ ” ~ (23) 9 ’
144 Lecourtier, Audebert, Quivoron
Macromolecules
It follows that P
b,l + N 3 In P7 + N2 In p”1
P
b,O
1
(25)
(L!
Lastly,
-ASM - - -NI In $I k
[
N3(?
- N 2 In $2 t - 1)
+ N 1 ] ln (1 -
N3(: - 1) ln (1 -
:)
3J3) +
[ *] ?$32/3)1’2
N31n
_ _
+
2 r3
3. Computing the Partition Coefficient KD. From the previous equations one finds the activity of the solute in the grafted phase and in the solvent. The following substitutions can be made assuming the solute is considered as very dilute: $2 (6’2 0 f$Jfl
Hence.
1
Discussion Partition coefficient KDis a function of the size of the three different kinds of molecules. I t largely depends on the relative magnitude of the interaction terms xi? If there is no enthalpic effect (x, = 0), KD varies from 0 to 1 as the molecular weight of the solute decreases from 00 to 0. Figure 3, in which the logarithm of the molecular weight of the solute ( M 2taken as 100 r2) is plotted versus KD, illustrates this prediction. The curves are similar t o the calibration curves of gel permeation c h r ~ m a t o g r a p h y . ~ ~ The solubility of high molecular weight solutes increases with the molecular weight of the grafted chains. The role of the solvent depends not only on the size of its molecules but mainly on its interactions with the solute and the gel (xlz and ~ 1 3 ) . When we consider a series of solvents all of which swell the grafted molecules identically (Le., the grafted molecules/solvent interactions are constant) but which have various affinities for the solute, the solubility of the solute into the grafted phase can be appreciably altered as shown on Figure 4. In this figure, curves for the logarithm of the molecular weight ( M , = 100r2)as a function of KD are plotted for the values xI3 = x 2 3 = 0,x12 = 0 (or 0.5), r3 = 10, and r3 = 500. On the other hand, as the affinity of the solute toward the grafted phase increases (Le., for decreasing values of x 2 3 ) , the curves log M 2 = f(KD) become warped and particularly when the solute/grafted chain interactions are very strong KD becomes greater than 1. Figure 5 shows the change for xi3 = 0.5, x12= 0, and r3 = 10.
Validity of the Proposed Model Our calculations are based on the lattice theory of polymer solutions developed by Flory13J4and by Huggins.17 The theory assumes that the interactions between the various species do not permit the formation of complexes which cannot easily be dissociated by thermal agitation a t the experimental temperature (hypothesis of “regular” systems). I t is also assumed that xi3 values are independent of 43,which is not always valid.I8 Several assumptions on the nature of the grafted phase have been made. The number of configurations has been calculated assuming r 3 / r l >> 1. The Surface bonded to and the grafted chains is equated with an inert plane, Le., it does not constrain segments of the grafted molecules to lie along it preferentially. As a general rule, we assume that no adsorption occurs a t the various interfaces of the system. In practice the validity of this assumption depends largely on the nature of grafts, surface, and For The condition a2 = u ‘ ~must be satisfied to arrive a t polystyrene deposit on silica, it has been shown experiequilibrium of the solute between the two phases. This that even in a 8 solvent, the greater part of the ~: equality gives the expression for the ratio In KD N ~ $ ~ / 4 ~ mentally
Swelling of Macromolecules Grafted Onto an Inert Surface 145
Vol. 12, No. I , January-February 1979
I og100i
log 1 OOf
4
3
2
1
4
0
0.5
Figure 3. log M2 (M2taken as 100r2)vs. KD (partition coefficient) for various values of r3. Curves plotted with rl = 1 and x I 2= xi3 = x23 = 0.
05 1 15 :K Figure 5. log Mz( M 2taken as 100rz)vs. KD(partition coefficient) for various values of x23. Curves plotted with r3 = 10 and x12 = X I 3 = 0.
is observed when adsorption of the macromolecule occurs on the surface. Smitham and NapperIg found t h a t the thickness of the attached layer corresponds to 2-2.5 times the unperturbed end-to-end distance of a polymer chain (Le., 2b(r3/ri)II2. Experimental results of Dorokowski and LambourneZ2lead to a thickness for the attached layer varying from 0.43b(r,/r,)1/2 to 2.7b(r3/r1)1/2. For our calculation we chose (Y b ( r 3 / r J 1 l 2 . The distribution of polymer segments into the grafted phase is assumed to be homogeneous. A factor expressing the correction for inhomogeneity of the solution can be introduced, but mathematical expressions would be considerably more c ~ m p l i c a t e d . ’ ~ ~ ~ The expressions proposed for the swelling and for the partition coefficient KDobtained for the grafted phases should be compared with those which were calculated for gels.24 Substituting the length between two cross-linkages of the gel for the length of the grafted molecules, the behavior of grafted materials and of gels appears to be similar. Hence, as for gels, our results should allow one to better understand the separation mechanisms on grafted phases involved in liquid-liquid chromatography. Such experimental studies have been performed in our laboratory. 15,25,26
References and Notes (1) F. R. Eirich, J . Colloid Interface Sci., 58, 432 (1977). (2) Yu S. Lipatov and L. M. Sergeeva, “Adsorption of Polymers”,
a5
KD
Figure 4. log M2 (M2taken as 100r2)vs. KD(partition coefficient) for various values of x I 2[(- - -1 x I 2= 0, (-1 x I 2= 0.51 plotted with r3 = 10 or 500 and x I 3 = x23 = 0.
segments of the adsorbed polymer molecules are far from the adsorbing surface and the thickness of the adsorbed layer varies like the radius of gyration of a macromolecule in s o l ~ t i o n . ’ ~ - ~ ~ T h e grafting of a macromolecular chain increases its extension perpendicular to the plane. The opposite effect
Wiley, New York, 1974. (3) P. G. De Gennes, J . Phys., 37, 144, 5 (1976). (4) E. F. Casassa, J . Phys. Chem., 75, 3929 (1971). (5) F. Verhoff and N. Sylvester, J . Macromol. Sci. Chem., 4, 979 (1970). (6) C. Guttmann and E. Dimarzio, Macromolecules, 3,681 (1970). (7) T. A. Weber and E. Helfand, Macromolecules, 9, 311 (1976). (8) D. J. Meier, J . Phys. Chem., 71, 1861 (1967). ( 9 ) F. Hesselink, J . Phys. Chem., 73, 3488 (1969). (10) M. Lax, Macromolecules, 7, 660 (1974). (11) E. J. Clayfield and E. C. Lumb, J . Colloid Interface Sci., 20, 285 (1966). (12) T. Tanaka, Macromolecules, 10, 51 (1977). (13) P. Flory, J . Chem. Phys., 10, 51 (1942).
Tanaka, Omoto, Inagaki
Macromolecules
P. Flory, “Principles of Polymer Chemistry”, Cornell University Press, Ithaca, N.Y., 1953. J. Lecourtier, These de Doctorat d’Etat, Universite P. et M. Curie, Paris, Dec. 19, 1977. E. F. Casassa, Sep. Sci., 6 (21, 305 (1971). M. Huggins, Ann. N.Y. Acad. Sci., 43, 1 (1942). P. Rempp in “Reactions on Polymers”, J. A. Moore, Ed., Dordrecht, Boston, 1973, p 265. J. B. Smitham and D. H. Napper, J. Polym. Sci., Polym. Symp., 55, 51 (1976).
R. Stromber, E. Passaglia, and D. J. Tutas, J . Res. Natl. Bur. Stand., Sect. A , 67, 431 (1963).
(21) G. Gebhard and E. Killmann, Angew. Makromol. Chem., 53, 71 (1976). (22) A. Doroszkowski and R. Lambourne, J. Colloid Interface Sci., 43, 97 (1973).
(23) E. Helfand, Polym. Prepr., Am. Chem. Soc., Diu. Polym. Chem., 15, 216 (1975). (24) J. Lecourtier, R. Audebert, and C. Quivoron, J . Chromatogr., 121, 173 (1976). (25) J. Lecourtier, R. Audebert, and C. Quivoron, J . Liq. Chromatogr., 1, 367 (1978). (26) J. Lecourtier, R. Audebert, and C. Quivoron, J. Liq. Chromatogr., 1, 479 (1978).
Conformation of Block Copolymers in Dilute Solution. 3. Determination of the Center-to-Center Distance between the Two Blocks by Light Scattering1 T a k e s h i T a n a k a , * M i t s u r u Omoto, a n d Hiroshi I n a g a k i Institute for Chemical Research, Kyoto University, Uji,Kyoto-Fu 611, Japan. Received October 30, 1978
ABSTRACT: The conformation of an AB diblock copolymer in dilute solution may be characterized by the mean-square radii ( S 2 ) A and ( S 2 ) B of the two blocks and the mean-square distance ( G ’ ) between the centers of mass of them. It was previously established that (S2)K (K = A or B) is almost identical with the radius ( S 2 ) H - K of the equivalent K homopolymer, unless the A-B interactions are attractive. In such a situation, it is possible to determine ( G2) by light scattering with a solvent having large refractive index increments for the two homopolymers. Experiments were conducted on polystyrene-poly(methy1methacrylate) diblock copolymers in 2-butanone. The parameter u representing the extent of “segregation” of the two blocks, u = (G2)/(2(S’), + 2(s2)B), was evaluated with due regard to the sample heterogeneity. The influence of heterogeneity was found to be unexpectedly large even for a fairly homogeneous sample, say M J M , < 1.1, and without adequate correction to this effect, the conclusion drawn from such an analysis could be misleading. It was found that g lies very close to 1.2. This shows that the conformation of the block copolymer is almost the same as that of the homopolymers. Models like “segregated”and “core-in-shell’’conformations are utterly at variance with reality. Despite the enormous effort so far made to understand the chain conformation of block copolymers in dilute solution, the progress until recently has been highly unsatisfactory.2a This is due, on the one hand, to crucial difficulties of dealing with the problem theoreticallyzband, on the other hand, to lack of effective experimental techniques which provide unequivocal information. This situation has given rise to considerable confusion regarding the simplest picture of block copolymer conformation, and thus various models, typified by the “quasi-random coil”, “segregated’, and “core-in-shell” conformations, have been proposed, without clear definitions for them.l Very recently, considerable success has been gained by computer “ e ~ p e r i r n e n t s ” .One ~ ~ of the important conclusions drawn thereby3 was that the mean-square radius ( S z ) K (K = A or B) of the individual block in an AB diblock copolymer is almost the same as the radius ( S 2 ) H - K of the equivalent K homopolymer, insofar as the A-B interactions are assumed to be nonattractive. More precisely, it was shown that the ratio YK given by ranges from 1.00 to 1.02 in a common good or 8 solvent for the two homopolymer^.^ Existing Monte-Carlo data on homopolymers3 and those on block copolymers recently reported by Bendler et al.5 are quite consistent with this conclusion. Interestingly, an artificial “segregated” chain which was simulated by a random walk biased by the presence of an impermeable, noninteracting plane passing through the A-B junction was found to give a value of YK = 1.03.8 0024-9297/79/2212-0146$01.00/0
The parameter Y K can be unambigously determined by light-scattering experiments: For a homogeneous diblock copolymer, the light-scattering apparent radius ( is given by (S2),pp
= I L A ( ~ ’ ) A + I L B ( ~ ’ ) B + ILAPB(G’)
PA =
1 - ILB = X A ~ A / ( X A ~ A+ XBVB)
(2)
where ( G 2 ) is the mean-square distance between the centers of mass of the two blocks, and X K and V K are the weight fraction and the refractive index increment of the K block, respectively. With a solvent in which vB = 0, for example, we have ( 9 ) A ( ~ ( p )Obviously, ~ ~ ~ this ) . method is applicable also to a heterogeneous block copolymer, for which the z-average radius of the A block is obtained. An analysis of existing light-scattering data of this kind led to the conclusion that Y K is unity within an experimental uncertainty of, say, f l O % .3 More recent neutron-scattering data of Benoit et al.9J0appear to be consistent with this conclusion. As far as we are aware, the only counterevidence for this is found in the paper of Han and Mozer.lla However, these experiments seem to leave room for criticism.llb For all these reasons, it is firmly believed that YK is, in a practical sense, equal to unity for a block copolymer showing repulsive interactions between the two blocks. Under such a situation, eq 2 may be well approximated by ( s ’ ) a p p = P*(S’)H-A + I L B ( S ’ ) H - B + ILAILB(G’) (3) With independently determined values of ( S’)H-K, we can 0 1979 American Chemical Society
